Effect of Thermal Dispersion and Thermal Radiation on Boundary Payer Flow of Mhd Nanofluid With Variable Suction

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1 IOSR Journal o Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 6 Ver. III (Nov. - Dec.6), PP Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid With Variable Suction K Uma maheswara rao and G koteswara Rao Department o Mathematics, Hyderabad Institute o Technology and management, Gowdavelly village, Rangareddy district, Hyderabad, Telangana 5 Department o Mathematics, Achaya Nagarjuna University, Nagarjuna Nagar, Guntur Abstract: The present study investigates the unsteady, two dimensional, laminar low o a nanoluid over a stretching sheet with thermal dispersion in the presence o MHD and variable suction is studied. Two water - based nanoluids containing Copper Cu, Titanium dioxide TiO nanoparticles are considered in this study. Using the similarity transormations, the governing equations have been transormed into a system o ordinary dierential equations. These dierential equations are highly nonlinear which cannot be solved analytically. Thereore, Runge Kutta Gill method together with shooting technique has been used or solving it. Numerical results are obtained or the skin-riction coeicient and the local Nusselt number as well as the velocity and temperature proiles or dierent values o the governing parameters, namely, unsteadiness parameter, solid volume raction, magnetic parameter, suction parameter, thermal dispersion parameter, thermal radiation parameter and Prandtl number. Keywords: Nanoluid, thermal dispersion, suction, MHD, thermal radiation. I. Introduction The development o a boundary layer over a stretching sheet was irst studied by Crane [], who ound an exact solution or the low ield. This problem was then extended by Gupta and Gupta [] to a permeable surace. Salleh et al. [3] studied the boundary layer low and heat transer analysis past a stretching sheet in the presence o Newtonian heating using inite dierence method. Variable thickness on boundary layer low over a stretching sheet is studied by Fang et al.[]. Khader and Megahed [5] both are studied the eects o variable thickness and slip velocity on boundary layer low towards a stretching sheet. Magneto-hydrodynamic (MHD) boundary layers with heat and mass transer over lat suraces are ound in many engineering and geophysical applications such as geothermal reservoirs, thermal insulation, enhanced oil recovery, packed-bed catalytic reactors, cooling o nuclear reactors. Many chemical engineering processes like metallurgical and polymer extrusion processes involve cooling o a molten liquid being stretched into a cooling system. The luid mechanical properties o the penultimate product depend mainly on the cooling liquid used and the rate o stretching. Some polymer liquids like polyethylene oxide and polyisobutylene solution in cetane, having better electromagnetic properties are normally used as cooling liquid as their low can be regulated by external magnetic ields in order to improve the quality o the inal product. A comprehensive review on the subject to the above problem has been made by many researchers (Yang et al., [6]; Trevisan and Bejan., [7]; Sparrow et al., [8]; Evans, [9]). Gangadbhar et al. [] investigated the eect i hydromagnetic and chemical reaction on heat and mass transer through a vertical plate with convective condition. Gangadhar and Baskar Reddy [] studied the similarity solution o magneto hydrodynamic boundary layer low o heat and mass transer through a moving vertical plate in a porous medium with suction. They concluded that magnetohydrodynamic eect signiicantly decreases the velocity boundary layer thickness. Gangadhar [] investigated the hydro magnetic boundary layer low o heat and mass transer with soret and duour eects. Rushi Kumar and Gangadhar [3] studied the eects o variable suction and heat generation eects on MHD boundary layer low o a moving vertical plate. They concluded that magnetic parameter reduces the velocity proiles. Rushi Kumar and Gangadhar [] investigated the ree convection boundary layer low between two parallel porous walls with MHD eect. In recent years tremendous eort has been given to the study o nanoluids. The word nanoluid coined by Choi [5] describes a liquid suspension containing ultra ine particles (diameter less than 5 nm). Experimental studies (e.g., Masuda et al. [6], Das et al. [7], Xuan and Li [8]) showed that even with a small volumetric raction o nanoparticles (usually less than 5%), the thermal conductivity o the base liquid is enhanced by -5% with a remarkable improvement in the convective heat transer coeicient. The literature on nanoluids was reviewed by Trisaksri and Wongwises [9], Wang and Mujumdar [] among several others. DOI:.979/ Page

2 Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. Bachok et al. [] investigated the dual solutions on boundary layer low o nanoluid over a moving surace in a lowing luid. The study o radiation eects has important applications in engineering. Thermal radiation eect plays a signiicant role in controlling heat transer process in polymer processing industry. Many studies have been reported on low and heat transer over a stretched surace in the presence o radiation (see El-Aziz [], Raptis [3], Mahmoud []). The eect o thermal radiation and MHD boundary layer low o the Blasius and Sakiadis lows with heat generation and viscous dissipation is studied by Gangadhar [5]. In another study, Gangadhar [6] investigated the laminar boundary layer low o nanoluid past a vertical plate in the presence o radiation and viscous dissipation. He concluded that solid volume raction increases the thermal boundary layer thickness. The eect o radiation and chemical reaction on MHD oscillatory low in a channel illed with porous medium is studied by Ibrahim et al.[7] Suneetha and Gangadhar [8] studied the similarity solution or MHD boundary layer low o a Carreau luid with thermal radiation eect. They concluded that Nusselt number reduces with an increasing the radiation parameter. Kameswaran et al. [9] studied the eects o radiation and thermal dispersion eects on boundary layer low o nanoluid. They considered water based copper oxide CuO, Aluminium oxide Al O 3 and titanium dioxide TiO nanoparticles. The present study investigates the unsteady, two dimensional, laminar low o a nanoluid over a stretching sheet with thermal dispersion in the presence o MHD and variable suction is studied. Two water - based nanoluids containing Copper Cu, Titanium dioxide TiO nanoparticles are considered in this study. Using the similarity transormations, the governing equations have been transormed into a set o ordinary dierential equations, which are nonlinear and cannot be solved analytically, thereore, Runge Kutta Gill method together with shooting technique has been used or solving it. The results or velocity and temperature unctions are carried out or the wide range o important parameters namely; namely, unsteadiness parameter, magnetic parameter, suction parameter, thermal dispersion parameter, thermal radiation parameter and Prandtl number. The skin riction coeicient and the rate o heat transer have also been computed. II. Mathematical Formulation Consider the unsteady two-dimensional boundary layer low over a stretching sheet in an electrically conducting water-based nanoluid containing dierent type o nanoparticles: copper ( Cu ) and titanium dioxide ( TiO ). The origin o the system is located at the slit rom which the sheet is drawn, with x and y denoting coordinates along and normal to the sheet. The thermo-physical properties o the nanoluid are speciied in Table-. Table : Thermo-physical properties oater and nanoparticles (Oztop and Abu-Nada [3]) Physical Water/base luid Cu TiO properties (kg/m 3 ) c p (J/kg K) k (w/m K) (S/m) The sheet is stretched with velocity bx Uw( x, t) (.) t Along the x axis, b and α are positive constants with dimensions (time) - and αt<. The surace temperature distribution r bx m Tw ( x, t) T T t (.) Various both along the sheet and with time, where T reerence temperature, T is the ambient temperature and is the kinematic viscosity o the luid. A uniorm transverse magnetic ield o strength B is applied parallel to the y -axis. It is assumed that induced magnetic ield produced by the luid motion is negligible in comparison with the applied one so that we consider the magnetic ield B,, B. This assumption is justiied, since the magnetic Reynolds number is very small or metallic liquids and partially ionized luids (Cramer and Pai [3]). Also, no external electric ield is applied such that the eect o polarization o luid is DOI:.979/ Page

3 Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. negligible (Cramer and Pai []), so we assume E,,. Under these assumptions, the boundary layer equations governing the low, heat and concentration ields can be written in a dimensional orm as ollows u v (.3) x y n n u u u n u n B u v u t x y y T T T T qr u v T n e t x y y c y y y p n The suitable boundary conditions are y : u U ( x, t), v v, T T w w w y : u, T T (.6) where u and v are the velocity components along the x and y axes, respectively, v w is the mass transer velocity, T is the temperature o the nanoluid, T is the ambient temperature, c is the speciic heat at constant pressure, r and m are constants, the expression or the eective thermal diusivity taken as du, is the molecular thermal diusivity, du represent thermal diusivity, is the mechanical e m m thermal dispersion coeicient and d is the pore diameter, thermal diusivity o the nanoluid and cp cp cp n Nada [3]). kn n, n s, n C p n DOI:.979/ Page p (.) (.5) n is the viscosity o the nanoluid, n is the is the density o the nanoluid, which are given by (Oztop and Abu-.5 ks k k ks kn, k k k k k n s 3( ) s n, s s Here, is the nanoparticle volume raction, c p is the heat capacity o the nanoluid, n is the electrical conductivity o the base luid, s is the electrical conductivity o the nanoparticle, k n is the thermal conductivity o the nanoluid, (.7) k and k s are the thermal conductivities o the luid and o the solid ractions, respectively, and and s are the densities o the luid and o the solid ractions, respectively. It should be mentioned that the use o the above expression or kn is restricted to spherical nanoparticles where it does not account or other shapes o nanoparticles (Oztop and Abu-Nada [3]). Also, the viscosity o the nanoluid has been approximated as viscosity o a base luid (Brinkman [3]). Following Rosseland s approximation, and the radiative heat lux q r is modeled as 3 qr 6 * T T y 3 k* y Where m is the Stean Boltzman constant and n containing dilute suspension o ine spherical particles T is expressed as a linear unction o the temperature 3 3 T T T T (.8) m is the mean absorption coeicient. The continuity equation (.3) is satisied by the Cauchy-Riemann equations u y and v x (.9) where ( xy, ) is the stream unction.

4 Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. the ollowing similarity transormations and dimensionless variables are introduced. / / b b bx ( x, y, t) x, y, u ( ) t ( t) t (.) / b T T v ( ), t Tw T where is the similarity variable, is the kinematic viscosity o the luid raction and b is a constant. Ater the substitution o these transormations (.7) - (.) into the equations (.) - (.6), the resulting nonlinear ordinary dierential equations are written as.5 ''' A '' ' S ''( ) '( ) ( ) Ha '( ) (.) k kn R '' D '( ) D ''( ) '( ) Pr ( ) '( ) r '( ) ( ) S '( ) m ( ) A k where kr 3 Nr.5 s A c p A s cp 3 A3 Together with the boundary conditions,, w, as (.3) (.) Here primes denote dierentiation with respect to. S is the unsteadiness parameter, D is the thermal dispersion parameter, Nr thermal radiation parameter, w is the suction parameter and w corresponds to injection, Pr is the Prandtl number, M is the magnetic parameter which are given by du k k * b c w S, D, N, v,pr b t k B M b m n p R 3 w w * T The physical quantities o interest are the skin riction coeicient (.) C, the couple stress C n and the local Nusselt number Nu x, which are deined as w xqw C Nu x u k (.5) w, T where the surace shear stress w and the surace heat lux qw are given by DOI:.979/ Page

5 Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. 3/ u / b w n ''() / x y y ( ) t / (.6) T b qw kn kn ( Tw T ) '() y t y with n and k n being the dynamic viscosity and thermal conductivity o the nanoluids, respectively. Using the similarity variables (.7)&(.), we obtain C x Nu Re x / x /.5 x Re ( ) ''() k k where Re n x ' Uwx is the local Reynolds number. (.7) III. Solution o The Problem For solving Eqs. (.) (.3), a step by step integration method i.e. Runge Kutta method has been applied. For carrying in the numerical integration, the equations are reduced to a set o irst order dierential equation. For perorming this we make the ollowing substitutions: y, y, y, y h, y h, y, y y5' Dy3 y k k R n Dy A k y (), y (), y () y3' A y y3 w y ( ), y 3 y ( ) S y 3 y ( ) Pr y y r y y A Hay S y 5 my In order to carry out the step by step integration o Eqs. Respseqn.-.3, Gills procedures as given in Ralston and Wil [33] have been used. To start the integration it is necessary to provide all the values o y, y, y, y at rom which point, the orward integration has been carried out but rom the boundary 3 conditions it is seen that the values o y3, y5are not known. So we are to provide such values o y3, y5along with the known values o the other unction at as would satisy the boundary conditions as to a prescribed accuracy ater step by step integrations are perormed. Since the values o y, y which are supplied are merely rough values, some corrections have to be made in these values in order 3 5 that the boundary conditions to are satisied. These corrections in the values o y3, y5are taken care o by a sel-iterative procedure which can or convenience be called Corrective procedure. This procedure has been taken care o by the sotware which has been used to implement R K method with shooting technique. Oh ; the global error would be Oh 5 As regards the error, local error or the th order R K method is. The method is computationally more eicient than the other methods. In our work, the step size h.. Thereore, the accuracy o computation and the convergence criteria are evident. By reducing the step size better result is not expected due to more computational steps vis-a` -vis accumulation o error. DOI:.979/ Page

6 Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. IV. Results And Discussion The governing equations (.) & (.) subject to the boundary conditions (.3) are integrated as described in section 3. In order to get a clear insight o the physical problem, the velocity and temperature have been discussed by assigning numerical values to the parameters encountered in the problem. The variation o velocity and temperature proiles or dierent values o magnetic parameter Ha is graphed in igures and or both Cu water nanoluid and TiO water nanoluid. From the graphs, in is seen that velocity proile signiicantly decreases with an increase in magnetic parameter Ha. As magnetic parameter increases, the Lorentz orce, which opposes the low, and increases and leads to, enhanced deceleration o the low. But the temperature proile increases with the increase in magnetic parameter. For the velocity distribution TiO water nanoluid is higher than that o Cu-water nanoluid. Under consideration o temperature distribution Cu-water nanoluid is higher than that o TiO water nanoluid because thermal conductivity o the Cu-water nanoluid is higher than that o TiO water nanoluid. Figures 3 and are graphed or dierent values o solid volume raction on the velocity and temperature proiles or both Cu-water nanoluid and TiO water nanoluid. From these igures, it is observed that both velocity and temperature distributions increases with an increase in solid volume raction. Figures 5 and 6 depict the variation o suction parameter on velocity and temperature proiles or both nanoluids. It is clearly observed that both velocity and temperature proiles signiicantly reduce with an increasing the values o suction parameter. Suction will leads ast cooling o the surace. Figures 7 and 8 is graphed or velocity and temperature proiles or dierent values unsteadiness parameter S. it is noticed that velocity proile decrease whereas temperature proile signiicantly reduces with the rising the values o unsteadiness parameter S. The variation in thermal dispersion parameter D on temeperature distribution or both Cu-water nanoluid and TiO water nanoluid is shown in igure 9. It is noticed that thermal dispersion increases, the temperature proiles increases. Figure depicts the eect o thermal radiation parameter Nr on temperature proiles or both nanoluids. It is observed that thermal boundary layer thickness increase with the inluence o thermal radiation. Figure is plotted or dierent values o unsteadiness parameter S, suction parameter, magnetic parameter Ha and solid volume raction on skin riction coeicient or both Cu water nanoluid and TiO water nanoluid. From the igure, it is noticed that skin riction coeicient signiicantly increase or increase in S, and Ha, whereas skin riction reduces with the rise in. The local skin riction coeicient is Cu water nanoluid higher than that o TiO water nanoluid. Figure is graphed or dierent values o unsteadiness parameter S, suction parameter, magnetic parameter Ha and solid volume raction on local Nusselt number or both Cu water nanoluid and TiO water nanoluid. From the igure, it is noticed that local Nusselt number signiicantly increase or increase in, whereas local Nusselt number reduces with the rise in S, Ha and. The local Nusselt number is higher TiO water nanoluid than that o Cu water nanoluid. The variation in thermal dispersion parameter D and thermal radiation parameter Nr on local Nusselt number or both nanoluids is shown igure 3. It is clearly observed that local Nusselt number reduces with an increase in both D and Nr. Table shows the very good agreement with previously published results. V. Conclusions In the present paper, unsteady, two dimensional, laminar low o a nanoluid over a stretching sheet with thermal dispersion in the presence o MHD and variable suction is studied. Two water - based nanoluids containing copper Cu, Titanium dioxide TiO nanoparticles are considered in this study. The governing equations are approximated to a system o non-linear ordinary dierential equations by similarity transormation. Numerical calculations are carried out or various values o the dimensionless parameters o the problem. It has been ound that velocity proiles decrease but temperature proiles increases with the inluence o magnetic ield. For the velocity distribution TiO water nanoluid is higher than that o Cu-water nanoluid. Under consideration o temperature distribution Cu-water nanoluid is higher than that o TiO water nanoluid. Both velocity and temperature proiles increases with an increase in solid volume raction. Both the thermal dispersion and thermal radiation increases the thermal boundary layer thickness. The local skin riction coeicient is Cu water nanoluid higher than that o TiO water nanoluid. The local Nusselt number is higher TiO water nanoluid than that o Cu water nanoluid. The local Nusselt number decreases with the inluence o thermal radiation and thermal dispersion. DOI:.979/ Page

7 ' () ' () Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid Dashed line : TiO Ha =.,, 5, r = m =.5 D = Nr = =.3 =.5 Pr = 6. S = Fig. Velocity distribution or various values o Ha ().6.. Ha =.,, 5, r = Dashed line : TiO m =.5 D = S =. =.3 Pr = 6. =.5 S = Fig. Temperature distribution or various values o Ha.6. Dashed line : TiO =.3,.,.5,.6 r = m =.5 Ha = D = Nr = S =. Pr = 6. = Fig.3 Velocity distribution or various values o DOI:.979/ Page

8 ' () Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. ().6.. Dashed line : TiO =.3,.,.5,.6 r = m =.5 Ha = D = Nr = S =. Pr = 6. = Fig. Temperature distribution or various values o..6.. Dashed line : TiO =,.5,, 3 r = m =.5 D = Nr = =.3 Ha = Pr = 6. S = Fig.5 Velocity distribution or various values o. ().6.. Dashed line : TiO =,.5,, 3 r = m =.5 D = S =. =.3 Pr = 6. Ha = S =. 6 8 Fig.6 Temperature distribution or various values o. DOI:.979/ Page

9 ' () Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid...6. Dashed line : TiO S =.,.,.,.6 r = m =.5 Ha = D = Nr = =.3 Pr = 6. = Fig.7 Velocity distribution or dierent values o S. ().6. r = Dashed line : TiO m =.5 Ha = D = Nr = =.3 Pr = 6. =.5. S =.,.,., Fig.8 Temperature distribution or various values o S. ().6.. Dashed line : TiO D =.,,, 6 r = m =.5 S =. Nr = =.3 Pr = 6. = Fig.9 Temperature distribution or dierent values o D. DOI:.979/ Page

10 Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. ().6.. Nr =.,.5,,.5 r = Dashed line : TiO m =.5 D = S =. =.3 Pr = 6. = Fig. Temperature distribution or dierent values o Nr. C x... S =,.5, Pr = 6., D=Nr=, r =, m=.5 =, Ha = =, Ha = =, Ha = =, Ha = =, Ha = =, Ha = Cu-Water TiO -Water Fig. local Skin-riction coeicient or various values o S,, Ha &. Nu x S =,.5, Pr = 6., D=Nr=, r =, m=.5 =, Ha = =, Ha = Cu-Water =, Ha = =, Ha = =, Ha = TiO -Water =, Ha = Fig. Local Nusselt number or dierent values o S,, Ha &. DOI:.979/ Page

11 Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. Fig.3 Local Nusselt number or dierent values o Nr & D. Table. Comparison or the values o () or the values o Pr, S when r =, m =.5, Ha = D = Nr =. Pr S ( ) Present study El-Aziz [33] Reerences []. Crane LJ, (97), Flow past a stretching plate, Z. Angew. Math. Phys., Vol., pp []. Gupta, PS, Gupta, AS, (977), Heat and mass transer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 55, [3]. Salleh MZ, Nazar R, Pop I., (), Boundary layer low and heat transer over a stretching sheet with Newtonian heating, Journal o the Taiwan Institute o Chemical Engineers, Vol.(6), pp []. Fang T, Zhang J, Zhong Y., (), Boundary layer low over a stretching sheet with variable thickness, Applied Mathematics and Computation, Vol.8 (3), pp [5]. Khader, M.M. and Megahed, A.M., (5), Boundary layer low due to a stretching sheet with a variable thickness and slip velocity, Journal o Applied Mechanics and Technical Physics, Vol. 56 (), pp 7. [6]. Yang J., Jeng D.R., and Dewitt K.J., (98), Laminar ree convection rom a vertical plate with Non-uniorm surace conditions, Number.Heat Transer, Vol.5, pp [7]. Trevisan O.V., Bejan A., (99), Combined heat and mass transer by natural convection in a porous medium, Adv.Heat Transer, Vol., pp [8]. Sparrow EM, Eckert ER and Minkowycz WJ, (96), Transpiration cooling in a magnetohydrodynamic stagnation point low, Appl.Sci.Res.A, Vol., pp.5-7. [9]. Evans HL., (96), Mass transer through laminar boundary layers, Int.J. Heat Mass Transer, Vol.5, pp []. Gangadhar K., Bhaskar reddy N., Kameswaran P.K., (), Similarity Solution o Hydro Magnetic Heat and Mass Transer over a Vertical Plate with a Convective Surace Boundary Condition and Chemical Reaction, International Journal o Nonlinear Science, Vol.3. No.3, pp []. Gangadhar, K., and Bhaskar Reddy, N., (3) Chemically Reacting MHD Boundary Layer Flow o Heat and Mass Transer over a Moving Vertical Plate in a Porous Medium with Suction, Journal o Applied Fluid Mechanics, Vol. 6, No., pp. 7-. []. Gangadhar, K., (3), Soret and Duour Eects on Hydro Magnetic Heat and Mass Transer over a Vertical Plate with a Convective Surace Boundary Condition and Chemical Reaction, Journal o Applied Fluid Mechanics, Vol. 6, No., pp [3]. Rushi Kumar B. and Gangadhar K., (), heat generation eects on mhd boundary layer low o a moving vertical plate with suction, Journal o naval architecture and marine engineering, Vol.9, pp DOI:.979/ Page

12 Eect o Thermal Dispersion and Thermal Radiation on Boundary Payer Flow o Mhd Nanoluid.. []. Rushikumar B., Gangadhar K., (), mhd ree convection low between two parallel porous walls with varying temperature, annals o aculty engineering hunedoara International Journal O Engineering Vol.3, ISSN pp.67-7 [5]. Choi, SUS, (995), Enhancing thermal conductivity o luids with nanoparticles. In: Proceedings o the ASME International Mechanical Engineering Congress and Exposition, pp ASME FED3/MD66, San Francisco, USA. [6]. Masuda, H, Ebata, A, Teramae, K, Hishinuma, N., (993), Alteration o thermal conductivity and viscosity o liquid by dispersing ultra-ine particles. Dispersion o Al O 3, SiO and TiO Ultra-Fine Particles. Netsu Bussei., Vol., pp [7]. Das, S, Putra, N, Thiesen, P, Roetzel, W., (3), Temperature dependence o thermal conductivity enhancement or nanoluids, J. Heat Trans., Vol.5, pp [8]. Xuan, Y, Li, Q, (3), Investigation on convective heat transer and low eatures o nanoluids, J. Heat Trans., Vol.5, pp [9]. Trisaksri, V, Wongwises, S., (7), Critical review o heat transer characteristics o nanoluids. Renew. Sustain. Energy Rev., Vol., pp []. Wang, XQ, Mujumdar, AS., (7), Heat transer characteristics o nanoluids: a review. Int. J. Therm. Sci., Vol.6, pp.-9. []. Bachok N., Ishak A, Pop I., (), Boundary-layer low o nanoluids over a moving surace in a lowing luid, International Journal o Thermal Sciences, Vol. 9(9), pp []. El-Aziz, MA, (6), Thermal radiation eects on magnetohydrodynamic mixed convection low o a micropolar luid past a continuously moving semi-ininite plate or high temperature dierences, Acta Mech., Vol. 87, pp.3-7. [3]. Raptis A., (998), Flow o a micropolar luid past a continuously moving plate by the presence o radiation, Int. J. Heat Mass Trans., Vol., pp []. Mahmoud, MAA, (7), Thermal radiation eects on MHD low o a micropolar luid over a stretching surace with variable thermal conductivity, Physica A., Vol. 375, pp.-. [5]. Gangadhar, K., (5), Radiation, Heat Generation and Viscous Dissipation Eects on MHD Boundary Layer Flow or the Blasius and Sakiadis Flows with a Convective Surace Boundary Condition, Journal o Applied Fluid Mechanics, Vol. 8, No. 3, pp , 5. [6]. Gangadhar K., (6), Radiation and Viscous Dissipation Eects on Laminar Boundary Layer Flow Nanoluid over a Vertical Plate with a Convective Surace Boundary Condition with Suction, Journal o Applied Fluid Mechanics, IF: 88, Vol. 9, No., pp [7]. Mohammed Ibrahim S., Gangadhar K. and Bhaskar Reddy N., (5), Radiation and Mass Transer Eects on MHD Oscillatory Flow in a Channel Filled with Porous Medium in the Presence o Chemical Reaction, Journal o Applied Fluid Mechanics, IF: 88, Vol. 8, No. 3, pp [8]. Suneetha S., Gangadhar K., (5), Thermal Radiation Eect on MHD Stagnation Point Flow o a Carreau Fluid with Convective Boundary Condition, Open Science Journal o Mathematics and Application, Vol.3(5), pp.-7. [9]. Kameswaran PK, Sibanda P and Motsa SS, (3), A spectral relaxation method or thermal dispersion and radiation eects in a nanoluid low, Boundary Value Problems, 3:. [3]. Oztop HF, Abu-Nada E (8), Numerical study o natural convection in partially heated rectangular enclosures illed with nanoluids. International Journal o Heat and Fluid Flow Vol.9, pp [3]. Cramer KR, Pai SI., (973), Magnetoluid dynamics or engineers and applied physicists. McGraw-Hill, New York. [3]. Brinkman H.C., (95), The viscosity o concentrated suspensions and solutions, The Journal o Chemical Physics, Vol. :57. [33]. Ralston, Wil, (96), Mathematical Methods or Digital Computers, John Wiley and Sons, N.Y., 7. [3]. El-Aziz M.A., (9), Radiation eect on the low and heat transer over an unsteady stretching sheet, Int. Commun. Heat Mass Trans., Vol.36, pp.5-5. DOI:.979/ Page